Galactic center GeV gamma-ray excess from dark matter with gauged lepton numbers

The recently observed excess in gamma-ray signal near the Galactic center suggests that dark matter particles may annihilate into charged fermions that produce gamma-ray to be observed. In this paper, we consider a leptonic dark matter, which annihilates into the standard model leptons, $\mu^+ \mu^-$ and $\tau^+ \tau^-$, by the interaction of the gauged lepton number ${\rm U(1)}_{L_\mu-L_\tau}$ and fits the observed excess. Interestingly, the necessary annihilation cross section for the observed gamma-ray flux provides a good fit to the value for the relic abundance of dark matter. We identify the preferred parameter space of the model after taking the existing experimental constraints from the precision measurements including the muon $(g-2)$, tau decay, neutrino trident production, dark matter direct detection, LHC, and LEP experiments.


I. INTRODUCTION
The dark matter (DM) problem is one of the pressing issues in particle physics and cosmology. While the existence of DM has been firmly established through various observations of its gravitational effects on multiple scales, its microscopic nature still remains unknown [1]. This situation stimulates a variety of DM searches including the direct detection of dark matter scattering off detector materials, the detection of indirect signals from the dark matter annihilation or decay, and the collider searches of missing energy signatures due to the produced dark matter particles. Of particular, we notice that the new cosmic-ray detection experiments, such as PAMELA [2], AMS-02 [3], and Fermi-LAT [4], based on satellites reach unprecedented sensitivity to the cosmic-ray signals, which leads better chance to get the indirect information of dark matter properties.
An intriguing observation was made using the public data of the Fermi Large Area Telescope (Fermi-LAT) by Hooper et. al. and also other independent groups [5][6][7][8][9][10][11][12][13][14][15][16]: a gammaray excess at E γ ≈ O(GeV) coming from the Galactic center (GC) is found. In the analyses, it is claimed that the gamma-ray excess spectrum is in good agreement with the emission expected from DM annihilation into standard model (SM) charged particles. 1 The GeV excess is well fitted by a DM particle with a mass of m DM ≈ 30 − 40 GeV annihilating to bb with an annihilation cross section of σv ≈ 2 × 10 −26 cm 3 /s [13,16]. 2 Silk et. al. pointed out that contributions of the diffuse photon emissions from primary and secondary electrons produced in DM annihilation processes are significant, especially for leptonic final states ( ¯ ) [14]. It is also noticed that with the inverse Compton scattering (ICS) and bremsstrahlung contributions from electrons, annihilations of DM particles with a mass of m DM ≈ 10 GeV into ¯ provide a good fit with an annihilation cross section of σv ≈ (1 − 2) × 10 −26 cm 3 /s [14].
The bb final state may be understood by Higgs portal type DM models and studied by several authors [21][22][23][24] but a model for the leptonic explanation based on leptophilic DM is relatively less studied for the GeV excess. Here we explore a leptophilic model with the DM mass m DM ≈ 10 GeV.
In the heavier mass domain, M DM ∼ > 100 GeV, leptophilic DM models have attracted a 1 In Ref. [17], the authors proposed a new mechanism naturally inducing a continuum bump signature in cosmic γ-ray measurements even with a particle directly decay into two photons, introducing Energy Peak idea together with the postulate of a generic dark sector [18]. 2 We note that recent observation of AMS-02 [19] has started to exclude the χχ → bb dominant DM explanation of relic abundance [20].
lot of attention (see e.g. [25].) due to recent observation of excessive cosmic-ray positron fraction by the PAMELA, Fermi-LAT, and AMS-02 experiments, but lack of excess in the anti-proton fraction [26]. In building the leptophilic DM model, it is attractive to gauge the differences in lepton numbers: U(1) Le−Lµ , U(1) Lµ−Lτ , and U(1) Lτ −Le . These symmetries are anomaly free without extending the SM particle contents [27][28][29]. 3 Leptophilic DM models with a U(1) L i −L j gauge symmetry for the positron excess were studied in Refs. [30][31][32]. In our analysis, we take U(1) Lµ−Lτ symmetry for the GeV gamma-ray excess since models with U(1) Le−Lµ , and U(1) Lτ −Le are stringently limited by existing cosmic-ray positron measurements in low energy [33].
It should be also noticed that astrophysical uncertainty in gamma-rays from around the GC including modeling of background emission in the inner galaxy is still big. Moreover, millisecond pulsars [6-8, 10, 12, 34] and pions from the collision of cosmic-rays with gas [6][7][8]10] can contribute to the GeV scale gamma-ray and have been proposed as alternative explanations of the GeV gamma-ray excess even though the spectral shape from millisecond pulsars looks too soft at the sub-GeV energy range to account for the observed GeV excess spectrum [35]. Also the morphological feature of the observed excess is extended to more than ∼ 10 • from the GC beyond the boundary of the central stellar cluster which could include numerous millisecond pulsars [13], and observed distributions of gas seem to provide a poor fit to the spatial distribution of the signal [13,36,37].
The contents of the paper is as follows. In Section II, we explain the leptophilic DM model in detail and present dominant annihilation channels. The model parameter space for the observed DM thermal relic abundance and the GeV gamma-ray excess is clarified.
In Section III, we discuss the existing experimental constrains on the same parameter space, then conclude in Section IV.

A. The model
We extend the SM: • by extending the gauge symmetry with U(1) Lµ−Lτ , • by introducing a new Dirac fermion ψ, which is identified as dark matter.
The charge assignment for the SM fermions and the new fermion regarding the L µ − L τ symmetry is given in Table I. The muon-leptons and anti-tau leptons are (+1), tau-leptons and anti-muon leptons are (−1) and the new fermion has a charge Q ψ . We take a universal gauge coupling constant g for Z interactions. For the (spontaneously broken) extended gauge symmetry, we associate a new vector boson Z with an undetermined mass m Z . The model Lagrangian is written as follows: where Q ψ,f are U(1) Lµ−Lτ charges of the DM and a SM fermion f , respectively given in Table I. In our study, the DM mass m ψ is taken to be 10 GeV to fit the GeV excess as suggested in Ref. [14] (see also Ref. [38]).
The ψ particle is neutral under the SM gauge interactions but its presence is seen by The gauge interaction allows an early time thermal equilibrium with the SM particles and the standard freeze-out took place at T ∼ m ψ /20 through the the dominant annihilation channels: where = µ, τ . The corresponding Feynman diagrams are depicted in Fig. 1. The DM annihilation into a Z pair is kinematically allowed only when m ψ > m Z . The leading order DM annihilation cross sections are given by where = µ, τ, ν µ , and ν τ . The decay width of the Z boson is given by where θ is the unit step function.

B. Relic abundance
Taking the DM relic density 0.11 < Ω DM h 2 < 0.13 [39], we found the preferred parameter space for ψ dark matter in m Z − g plane for Q ψ = 2 in Figure 2. The plots for other values of Q ψ are also given later. The ballpark range is 1 < m Z [GeV] < 500 and 0.001 < g < 1.0 as a reasonable choice within the perturbative regime. Notably, the dip structure appears around m Z 2m ψ = 20 GeV due to the resonance in the s-channel annihilation into leptons is required to fit the GeV gamma-ray excess. mediated by the Z gauge boson. In calculating the thermal average of DM annihilation cross section for the relic abundance, we take the the non-negligible effect of DM kinetic energy near the resonance pole, m Z 2m ψ = 20 GeV as explained in Ref. [40]. The result is shown in Fig. 2.

C. Fermi-LAT GeV excess
We conduct the fit of our model, µ + µ − : τ + τ − = 1 : 1, to the observed spectrum of the GC GeV γ-ray excess. Our best fit is obtained for σv ψψ→µ Fits to the GC GeV γ-ray excess for 10 GeV DM annihilating into µ + µ − and τ + τ − with branching ratios of µ + µ − : τ + τ − = 1 : 1. The best fit is obtained with σv ≈ 1.22 × 10 −26 cm 3 /s, which is plotted as a red line. Upper and lower fits corresponding to a p-value greater than 10 −3 are presented as purple dashed and dotted curves, respectively. The black points with blue error bars are the data points extracted from Ref. [14].
black dots and their error bars are represented by blue lines. As can be seen in Fig. 2, the plot for the right relic abundance reproduces successful GeV excess in the GC as was originally observed in [5,6] and also in [14] for leptonic annihilations taking into account the contributions by the ICS and bremsstrahlung with the annihilation cross section of

III. EXPERIMENTAL CONSTRAINTS FOR THE PREFERRED PARAMETER SPACE
We now check whether the preferred parameter space m Z ∼ O(10 − 100) GeV and g < 1 is still available after taking the relevant experimental constraints from the processes potentially induced by the gauged lepton number interactions: (g − 2) µ , τ decay, neutrino trident production, loop-induced DM-nucleon scattering and leptonically interacting Z searches at colliders.
Thus, there exists discrepancy between the experimental value and the SM prediction: Even though the difference may be a sign of new physics but, more conservatively, we would set an upper bound on the size of the new contribution given in Eq. (7) and find the 2σ bound line in Fig. 8.

B. τ decay
The gauged lepton number interaction may be seen in the leptonic decay of tau through the box diagrams such as Fig. 4(b), which could make the branching fraction, Br(τ → µν τνµ ), larger than what the SM predicts. It is interesting to notice that the measured value of the tau decay branching fraction to µν τ ν µ is indeed slightly larger than what the SM predicted: the measured values for the branching ratio, Br(τ → µν τ ν µ ), and the life time of tau from the PDG [44] are Br(τ → µν τ ν µ )| PDG = (17.41 ± 0.04)%, which has more than 2σ deviation from the SM prediction [45,46]: From the box diagrams with the Z mediation, the deviation ∆ could be evaluated [46]: Interestingly, the sign of ∆ from the U(1) Lµ−Lτ interaction is consistent with that required by the difference between the experimental value and the SM prediction, Eq. (13). In Fig. 8, the upper region of an orange curve is excluded by the τ → µν τ ν µ decay limit at the 2σ level.

C. Neutrino trident production
Neutrino trident production, ν µ N → ν µ N µ + µ − , has been observed by several neutrino beam experiments such as CHARM-II [47] and CCFR [48]: The measured cross sections are consistent with the SM prediction so that stringently constrain our model. In the SM, the neutrino trident production is induced by a µ + µ − pair production from the scattering of a muon-neutrino in the Coulomb field of a target nucleus [46,49]. In our model, the leading order correction is coming from the contribution of Z boson shown in Fig. 5(a) that interferes with the SM contribution from similar diagrams with a W/Z boson exchange instead of the Z . In our analysis, we use the exclusion limit (95% C.L.) obtained from the CCFR data in Ref. [49] which is shown as a light cyan-shaded region with the cyan dot-dashed curves in Fig. 8.

D. Dark matter direct detection
DM direct detection experiments search for the recoil energy of nucleus by DM scattering off nucleus. In this model, DM does not directly couple to quarks at tree-level. However, one-loop suppressed scattering processes such as the one shown in Fig. 5(b) can still provide a sizable DM-nucleus scattering cross section in spite of the loop suppression factor. The one-loop suppressed DM-nucleus scattering cross section is given by [50] where Λ = m Z /(g Q ψ ) is the effective cut-off scale, µ N = m N m ψ /(m N + m ψ ) is the DM-nucleus reduced mass, and Z is the atomic number, i.e. the EM charge of the target nucleus. Note that Eq. (17) originally has a log dependence on the renormalization scale due to the fermion loop. However, such log dependences from µ-and τ -loops cancel each other out thanks to the relative sign difference between µ-and τ -loop induced diagrams.
In order to directly compare the DM-nucleus cross section with experimental bounds, we convert Eq. (17) into the DM-nucleon cross section using the following relation: where A is the atomic mass number of the target nucleus and µ n is the DM-nucleon reduced mass. For m ψ 10 GeV, the most stringent direct detection bound is currently provided by the LUX experiment [51]. The LUX limit is shown as a purple dashed line in Fig. 8.

E. Searches for Z → 4 at the LHC and LEP
The LHC results also provide constraints on the gauged lepton number interactions through the lepton productions. A single Z production is allowed at tree-level at hadron colliders such as the LHC in pp → + − Z where the Z boson is radiated from a lepton in the Drell-Yan process as shown in Fig. 6 even though Z interaction is lepton-specific. The produced Z boson subsequently decays either to a pair of charged-leptons, neutrinos or DM particles: if kinematically allowed. These processes can be probed by detecting either one chargedlepton pair plus missing E T events or two charged-lepton pairs, i.e. 4 , at the LHC. In this work, we focus on the 4 signals due to its clean and distinctive signature. If m Z (m τ , m ψ ), the branching ratios of the Z become − , ν , ψ 6. Feynman diagram for a Z boson production process at a hadron collider. The Z boson is radiated from a lepton, and then decays into a pair of leptons or DM's.
The leptophilic Z can be detected at the LHC in four charged-lepton final states. The dominant SM backgrounds for this process are Four charged-lepton (4 ) production at the Z resonance has been already measured by ATLAS [52] and CMS [53] collaborations at the LHC. Three final states have been well observed: pp → 4e, 2e2µ, 4µ. We consider only the four muon final state since in our scenario the Z does not couple to electrons. In this analysis, we use the following selection cuts which is used in the ATLAS analysis [52]: • P T,µ > 4 GeV and |η| < 2.7 for individual muons, • Separation of muons: ∆R µµ > 0.1, • Invariant mass of a muon pair: M µ + µ − > 5 GeV, • Invariant mass of four muons: 80 GeV < m 4µ < 100 GeV.
In Fig. 7, we present the Z production cross sections through the pp → µ − µ + Z process for g = 0.1 at the 8 and 14 TeV LHC which is obtained using MadGraph [54].
This Z can be produced at tree-level at lepton colliders such as LEP through the similar process as shown in Fig. 6 just by replacing qq with e + e − since the gauged U(1) Lµ−Lτ boson also has no direct coupling to e + and e − . The potential constraint from LEP for m Z < m Z has been well studied in Ref. [42] through the process, e + e − → µ + µ − Z . Despite much smaller total integrated luminosity, the limit from LEP is comparable to that from the 8 TeV LHC due to much cleaner signals. In Fig. 8, we present the LEP limit on Z from Ref. [42] as a dark-gray shaded region with the black long-dashed curves.
F. Summary of experimental constraints on the U(1) Lµ−Lτ model In Fig. 8, we collectively depict all the relevant constraints to the gauged lepton number interaction in m Z − g plane.
• The limit from (g − 2) µ : We plot the 2σ limit from (g − 2) µ as a green solid line in the m Z − g plane for representative choices Q ψ = 2, 1, 0.5, and 0.1. The upper region of the green line is constrained by the current measurements of (g − 2) µ .
• The limit from τ → µν τ ν µ decay: The upper region of the orange curve is excluded by the τ → µν τ ν µ decay limit at the 2σ level.
• The limit from Neutrino trident production: The 95% C.L. exclusion limit is shown as a light cyan-shaded region with a cyan dot-dashed curve. • The limit from dark matter direct detection: The LUX limit is plotted as purple dashed lines for four representative values Q ψ = 2, 1, 0.5, and 0.1.
• LHC Z → 4 limit: The light gray-shaded region with the gray dotted curve is excluded by measurements of the Z → 4µ at the LHC [49,50]. The Z → 4µ searches at the LHC strongly constrain the parameter space of m Z ≈ 5 − 40 GeV since the 4 production has been measured at the Z resonance and the selection cuts of P T,µ > 4 GeV and M µ + µ − > 5 GeV are used.
• LEP Z → 4 limit: Dark grad-shaded region with the black long-dashed curve is excluded by measurements of the Z → 4µ at LEP [42].
For Q ψ 1, considerable parameter space has already been ruled out by neutrino trident production and Z → 4µ observations at the 8 TeV LHC and LEP, except the region around the resonance of m Z ≈ 2m ψ . In near future, for larger Q ψ 1 most of preferred parameter space will be verified by DM direct detection experiments such as XENON1T. The region around the resonance will be complementarily proved by Z → 4µ searches at the 14 TeV LHC.

IV. CONCLUSION
In this work, we have explored a